• Projection onto the Lowest Landau level

Projection onto the Lowest Landau level

A convenient formulation of quantum mechanics within the subspace of the lowest Landau level wave developed by Girvin and Jach, and was exploited by Girvin, MacDondau, and Platzman in the magneto-roton theory of collective excitations of the incompressible states responsible for the fractional quantum Hall effect.

Briefly, we first consider the one-body case and choose the symmetric gauge. The single-particle eigenfunctions of kinetic energy and angular momentum in the LLL are

\[\phi_m=\frac{1}{(2\pi2^mm!)^{1/2}}z^m \exp(-|z^m|/4)\]

where \(m\) is a non-negative integer, and \(z=\frac{x+iy}{\ell}\). Clearly, any wave function in the LLL can be written in the form of an analytic function of $z$ times a Gaussian factor

\[\psi(z)=f(z)e^{-\frac{|z|^2}{4}}\]

So, the subspace in the LLL is isomorphic to the Hilbert space of analytic functions. Accordingly, we can define the inner product \((f,g)\) of two analytic functions \(f(z),g(z)\) with the measure

\[d\mu(z)\equiv (2\pi)^{-1}dxdye^{-|z|^2/2}.\]

We can define bosonic ladder operators that connect \(\phi_m\) to \(\phi_{m+1}\).

\[\begin{align} a^\dagger & =\frac{z}{\sqrt{2}}, \\ a & = \sqrt{2} \frac{\partial}{\partial z}, \end{align}\]

which satisfies the conditions

\[\begin{align} a^\dagger \varphi_m & = \sqrt{m+1}\varphi_{m+1}, \\ a \varphi_m & = \sqrt{m}\varphi_{m-1}, \\ (f,a^\dagger g) & =(af,g), \\ (f,ag) &=(a^\dagger f,g). \end{align}\]

All operators that have non-zero matrix elements only within the LLL can be expressed in terms of \(a\) and \(a^\dagger\). It is essential to notice that the adjoint \(a^\dagger\) is not \(z^*/\sqrt{2}\) but \(a= \sqrt{2}\partial/\partial z\) of \(z^*/\sqrt{2}\) onto the LLL by following

\[(f, \frac{z^*}{\sqrt{2}}g) = (\frac{z}{\sqrt{2}}f,g)=(a^\dagger f,g)=(f,ag).\]

So we find \(\bar {z^*} = 2\frac{\partial}{\partial z}\). (With \(\bar{}\) denoting the projected operator.)

• Since \(\bar{z}^*\) and \(z\) do not commute when we need to project an operator which is a combination of \(z^*\) and \(z\), we must first normal order \(z^*\) to the left of $z$ and then replace \(z^*\) with \(\bar z^*\).

With this rule in mind, we can easily project onto the LLL any operator that involves space coordinates only.

One example is the one-body density operator in momentum space

\[\begin{align} \rho_\mathbf q & = \frac{1}{\sqrt{A}}e^{-i\mathbf q\cdot \mathbf r} \\ & =\frac{1}{\sqrt{A}}e^{-\frac{i}{2}(q^*z+qz^*)} \\ & =\frac{1}{\sqrt{A}}e^{-\frac{i}{2}qz^*}e^{-\frac{i}{2}qz}, \end{align}\]

where \(A\) is the area of the system, and \(q = q_x+iq_y\). Hence

\[\begin{align} \bar \rho_q & = \frac{1}{\sqrt{A}} e^{-iq\frac{\partial }{\partial z}}e^{-\frac{i}{2}q^*z} \\ & = \frac{1}{\sqrt{A}}e^{-\frac{|q|^2}{4}}\tau_q, \end{align}\]

where \(\tau_q = e^{ -iq \frac{\partial}{\partial z}-\frac{i}{2}q^* z}\) is a unitary operator satisfying the closed Lie algerba.

\[\begin{align} \tau_q\tau_k & = \tau_{q+k}e^{\frac{i}{2} q\wedge k} \\ [\tau_q,\tau_k] & =2i\tau_{q+k}\sin\frac{q\wedge k }{2} \end{align}\]

Where \(q\wedge k=\ell^2 (\mathbf q\times \mathbf k )\cdot \hat {\mathbf z}\). We also have \(\tau_q\tau_k\tau_{-q}\tau_{-k}=e^{i q\wedge k }\). This is a familiar feature of the group of translations in a magnetic field because \(q\wedge k\) is exactly the phase generated by the flux in the parallelogram generated by \(\mathbf q \ell^2\) and \(\mathbf k\ell^2\). Hence, \(\tau\) forms a representation of the magnetic translation group. In fact, \(\tau_q\) translates the particle a distance \(\ell^2 \hat{\mathbf z}\times \mathbf q\) . This means that different wave vector components of the charge density do not commute. It is from here that non-trivial dynamics arise even though the kinetic energy is quenched in the LLL subspace.

This formalism is generalized to the case of the many particles with spins. In a system with area \(A\) and \(N\) particles the projected charge and spin density are

\[\begin{align} \bar{\rho_q} & = \frac{1}{\sqrt{A}} \sum_i^N \overline{e^{-i\mathbf q\cdot\mathbf r_i}} \\ & =\frac{1}{\sqrt{A}}\sum_i^N e^{-\frac{|q|^2}{4}}\tau_q(i)\\ \overline{S_q^\mu} & =\frac{1}{\sqrt{A}} \sum_{i=1}^N \overline{e^{-i \mathbf{q}^{\prime} \cdot \mathbf{r}_i}} S_i^\mu \\ & =\frac{1}{\sqrt{A}} \sum_{i=1}^N e^{-\frac{|q|^2}{4}} \tau_q(i) S_i^\mu, \end{align}\]

• We immediately find that, unlike the unprojected operators, the projected spin and charge density operators do not commute: \(\left[\bar{\rho}_k, \bar{S}_q^\mu\right] =\frac{2 i}{\sqrt{A}} e^{\frac{|k+q|^2-|k|^2-|q|^2}{4}} \overline{S_{k+q}^\mu} \sin \left(\frac{k \wedge q}{2}\right) \neq 0\) This implies that within the LLL, the dynamics of spin and charge are entangled, when you rotate spin, the charge gets moved. As a consequence of that, spin textures carry a charge.